Nori's structural-rank conjecture for closures of algebraic-group subgroups

Let $\mathcal{A}$ be a commutative algebraic group over $\mathbb{Q}$ that is either a torus or an abelian variety, with Lie algebra $\mathfrak{a}$ over $\mathbb{Q}$ and exponential map

$$\operatorname{exp}:\mathfrak{a}\otimes\mathbb{R}\longrightarrow\mathcal{A}(\mathbb{R}).$$

Let $\mathcal{L}\subset\mathcal{A}(\mathbb{Q})$ be finitely generated, torsion free, and of rank $m$. For each finite prime $p$ for which its closure $\overline{\mathcal{L}}_p$ in $\mathcal{A}(\mathbb{Q}_p)$ is compact and contains $\mathbb{Z}_p^{d(p)}$ as a finite-index subgroup, define $d(p)$ accordingly. Nori's structural-rank conjecture. The integers $d(p)$ are independent of $p$ and equal the structural rank of the linear map $\mathbb{Z}^m\to\mathfrak{a}\otimes\mathbb{R}$ induced by $\log\mathcal{L}$. This is proposed as a common-value refinement of Nori's question, with no resolution evidence supplied.